Fundamental Misunderstanding of SR…?

1. The problem statement, all variables and given/known data
A space explorer travels in a spaceship with v = 0.9c. She goes from Earth to a distant star that is 4 light years away (again, measured from Earth). What is the distance measured by the explorer and how long will she say it took her to get there?

3. The attempt at a solution
I solved the first part perfectly fine and got the answer of ##L = 1.7## light years. When solving for ##\Delta t## I realized that I didn't have ##\Delta t_0##. No problem! I just solved the equation ##\frac {L_0} v## for ##\Delta t_0## and got a value of ##\frac {40} {9}## years. Since ##\gamma \approx 2.3## I just multiplied ##2.3## by ##\frac {40} {9}## and got a value of ##10.2## years. The actual value is 1.9 years, which happens to also be ##\frac {\Delta t_0} {\gamma}## but I don't see why I'm wrong!

NOTE: I know that there is another solution to this problem and perhaps it is a bit simpler. I would like to focus on whats wrong with my thought process and fix THAT rather than try to adapt to another method of solving.